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21st June 2020, 07:55 AM  #1  
Penultimate Amazing
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Deeper than primes  Continuation 1/3*9
You wrote
The focus is only on A <= B proposition, which can be written also as (A < B) OR not(A < B) (a tautology), exactly because not(A < B) can't be but (A = B) in case of A <= B proposition. Cardinality is a measure of the number of members of set A (notated as A) Set A is called finite iff given any n in N, A is any particular n Set A is called nonfinite iff given any n in N, A is not any particular n By the standard notion "given any" is the same as "for all" ( as seen in https://en.wikipedia.org/wiki/Universal_quantification ) but not in my framework, where "give any" holds for both finite and nonfinite sets, where "for all" holds only for finite sets. Nonfinite sets have immediate or nonimmediate successors exactly because given any n in N, A is not any particular n. This is not the case with finite sets, they do not have immediate or nonimmediate successors exactly because given any n in N, A is any particular n.


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That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

21st June 2020, 08:42 AM  #2 
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No. I have defined what the expression, A <= B, means. It means there exists an injection from A to B.
I even provided meanings for A = B and for A < B. You can, in fact, use those meanings to show that the expression A <= B is identical to (A < B) OR (A = B), but not the nonsense you produced. The negation of A < B is definitely not A = B. 
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21st June 2020, 09:02 AM  #3 
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Fine. You've established a general notion and a notation.
Quote:
Set A is called nonfinite iff given any n in N, A is not any particular n[/quote] Ditto. So, still no definition for cardinality.
Quote:
But since you have, your whole argument has collapsed. You have nothing. Your criticisms of Mathematics vanish into a dust cloud of things you just make up. 
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22nd June 2020, 02:50 AM  #4 
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I reexamined our last discussions, and we are not "in the same page" about the meaning of "a relative measure of "sizes" of two sets".
This is "my page": A relative measure of "sizes" of two sets does not have to be done by specific directions (the terms "from A to B" or "from B to A" are irrelevant).  Let _ be a placeholder for any given set. Definition 1: Cardinality is the measure of the number of members of _ (notated as _) Definition 2: Cardinality between the two sets A and B is a relative measure of their numbers of members where ( ((A < B) iff (B > A)) OR ((A = B) iff (B = A)) ) (The meanings for equality (=) and inequalities (< or >) of cardinalities follow directly (also visually in "((A < B) iff (B > A))" there are two different symbols ("<" and ">"), where in "((A = B) iff (B = A))" there is the same symbol ("="), which intuitively reinforces the meaning of inequality or equality)). In simpler words, Cardinality is the measure of the number of members of sets A and B, where their possible relations are = or < or >. It says nothing about being finite or nonfinite set. So: Definition 3: Set A is called finite iff given any n in N, A is any particular n Definition 4: Set A is called nonfinite iff given any n in N, A is not any particular n  By the standard notion "given any" is the same as "for all" ( as seen in https://en.wikipedia.org/wiki/Universal_quantification ) but not in my framework, where "give any" holds for both finite and nonfinite sets, where "for all" holds only for finite sets. Nonfinite sets have immediate or nonimmediate successors exactly because given any n in N, A is not any particular n. This is not the case with finite sets, they do not have immediate or nonimmediate successors exactly because given any n in N, A is any particular n. 
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22nd June 2020, 05:16 AM  #5 
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22nd June 2020, 05:58 AM  #6 
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Definition 2 is not my definition of Cardinality, but uses Definition 1 in order to define Cardinality between two sets.
Thank you. I forgot to add the following in my previous post: "In simpler words, Cardinality between the two sets is the relative measure of the numbers of members of sets A and B, where their possible relations are = or < or >." 
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22nd June 2020, 06:34 AM  #7 
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Yes, but

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22nd June 2020, 09:12 AM  #8 
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22nd June 2020, 09:24 AM  #9 
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23rd June 2020, 07:47 AM  #10 
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Cardinality (by Definition 1) is defined as the measure of whatismeasured, where whatismeasured is "the number of members of _ (where _ is a placeholder for any given set (and it is (Cardinality) notated as _).
The information of how the measure is taken from whatismeasured, is provided by Definition 2.  Let's look at the standard definition of Cardinality: Cardinality is a relative measure of "size" of sets where A <= B iff there exists an injection from A to B." By the standard definition it is informed that A or B are stand for sets. Can you tell me please what A or B are stand for, by the standard definition? 
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23rd June 2020, 09:18 AM  #11 
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23rd June 2020, 09:31 AM  #12 
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"A measure of whatismeasured"  What a bizarre thing to write. Meaningless and bizarre, both.
"the number of members of [a set]"  not bizarre, but also not part of a definition. The axioms of set theory have no concept of "number of". You'd need to add that. In the socalled Definition 2, you use three undefined relationships (<, =, >). They have no meaning in set theory unless you define them. I can only assume you are assuming a meaning from arithmetic. If so, then all you've done is implied A has a numeric value and shown a couple of trivial properties of the relations. 
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23rd June 2020, 09:33 AM  #13 
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23rd June 2020, 10:38 PM  #14 
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Please this time answer to the following:
Let's look at the standard definition of Cardinality: Cardinality is a relative measure of "size" of sets where A <= B iff there exists an injection from A to B." By the standard definition it is informed that A or B stand for sets. Can you tell me please what A or B are stand for, by the standard definition? 
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24th June 2020, 04:07 AM  #15 
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Let _ be a placeholder for any given set.
Now, some corrections of definitions 1 and 2. Definition 1: Cardinality is the "size" of _ (notated as _), which is determined by the members of N, where 0 is a member of N. A relative measure of "sizes" of two sets does not have to be done by specific directions (the terms "from A to B" or "from B to A" are irrelevant). Let < be less than. Let > be greater than. Let = be equal to. Definition 2: The cardinality between the two sets A and B is a relative measure of their "sizes", where ( ((A < B) iff (B > A)) OR ((A = B) iff (B = A)) ) (visually in "((A < B) iff (B > A))" there are two different symbols ("<" and ">"), where in "((A = B) iff (B = A))" there is the same symbol ("="), which intuitively reinforces the meaning of inequality or equality). These definitions say nothing about being finite or nonfinite sets. So: Definition 3: Set A is called finite iff given any n in N, A is any particular n Definition 4: Set A is called nonfinite iff given any n in N, A is not any particular n  By the standard notion "given any" is the same as "for all" ( as seen in https://en.wikipedia.org/wiki/Universal_quantification ) but not in my framework, where "given any" holds for both finite and nonfinite sets, where "for all" holds only for finite sets. Nonfinite sets have immediate or nonimmediate successors exactly because given any n in N, A is not any particular n. This is not the case with finite sets, they do not have immediate or nonimmediate successors exactly because given any n in N, A is any particular n. Let ≤ be less than or equal to. Let ≥ be greater than or equal to. Definition 5: The cardinality between sets A and B is called nonstrict inequality iff ((A ≤ B) iff (B ≥ A)). 
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That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

24th June 2020, 05:03 AM  #16 
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24th June 2020, 05:05 AM  #17 
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24th June 2020, 05:14 AM  #18 
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I do not repeating my self in http://www.internationalskeptics.com...9&postcount=15 since it is a correction of http://www.internationalskeptics.com...03&postcount=4.

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24th June 2020, 05:28 AM  #19 
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That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

24th June 2020, 11:25 AM  #20 
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25th June 2020, 06:04 AM  #21 
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If A and B are arbitrary sets and . is a standard notation for cardinality, then your definition of Cardinality (which is for a relative measure, i.e., a comparison) is:
Definition of Cardinality: The cardinality of A <also notated as A> <= the cardinality of B <also notated as B> iff there is an injection from A to B. (Also you skipped on http://www.internationalskeptics.com...8&postcount=18). 
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25th June 2020, 10:38 AM  #22 
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No, it is not. The definition is for a specific relationship between cardinalities of sets. Your propensity for overstating the trivial and the bloody obvious notwithstanding, A <= B is what is being defined. Not A nor B.
That is the only way a relative measure can be defined, i.e., in terms of how measurements would compare. 
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28th June 2020, 05:09 AM  #23 
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A relative measure is defined iff what is compared and how it is compared, are both or neither taken.
In this case what is compared is the cardinalities of sets A and B (which are A and B), where how they are compared is done by onetoone function (also called injection, such that A <= B). Now look at this: If A and B are arbitrary sets and . is a standard notation for cardinality, then your definition of Injection (which is for a relative measure, i.e., a comparison) is: Definition of Injection : There is a onetoone function, also called injection, from A to B iff The cardinality of A <also notated as A> <= the cardinality of B <also notated as B> Definition of Cardinality: The cardinality of A <also notated as A> <= the cardinality of B <also notated as B> iff there is an injection from A to B.  (Also you skipped on http://www.internationalskeptics.com...8&postcount=18). 
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28th June 2020, 12:52 PM  #24 
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28th June 2020, 07:26 PM  #25 
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Not much of a correction, and you didn't address the objections raised.
Nonetheless... Or we could just follow convention and use a letter of the alphabet instead of a cryptic underbar.
Quote:
N? What is N? A set, I presume, but what set? 0? What is 0? Set theory has nothing like that. You'll need to define it.
Quote:
...and so on for the rest of the "corrected" post. You have not moved any closer to a definition. Please keep trying, though. 
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29th June 2020, 03:37 AM  #26 
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In other words, you have missed what is written in http://www.internationalskeptics.com...1&postcount=23.
Please read all of it, then think about it, and only then please air your view about its content. Thank you. 
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29th June 2020, 03:48 AM  #27 
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As long as you try to understand http://www.internationalskeptics.com...9&postcount=15 in terms of ZF(C), you are surly missing it.

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29th June 2020, 06:00 AM  #28 
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29th June 2020, 06:07 AM  #29 
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30th June 2020, 06:24 AM  #30 
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jsfisher, no relative measure is defined unless what is compared (and in this case it is the cardinality of set A (notated as A) and the cardinality of set B (notated as B)) and how it is compared (and in this case it is an injection from set A to set B), where the logical connective between what is compared and how it is compared is iff, otherwise A <= B can't be defined.
Any attempt to define X in terms of relative measure, can't avoid X as a part of the definition. And this is exactly what I show in http://www.internationalskeptics.com...1&postcount=23, which you get as gibberish as long as you are missing the truth table of iff where: In case of relative measure Code:
p="what is measured" q="how it is muasured" p iff q  F F T F T F T F F T T T 
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30th June 2020, 07:28 AM  #31 
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What is defined in http://www.internationalskeptics.com...9&postcount=15 is simpler than your definitions that are done in terms of ZF(C) (your definition of Cardinality, which uses cardinality as a part of the definition).
For example: 1) I do not need iff in order to define Cardinality by the members of the set of natural numbers (notated as N), where natural numbers (including number 0) are naturally understood (no extra maneuvers are needed). 2) I provide symbols to the concepts "less than", "equal to", "greater than" (which I tend to replace by "more than") which are naturally understood (no extra maneuvers are needed). 3) I use iff in a vary simple way in case of relative measure, between two arbitrary sets A and B, where their cardinalities (what is compared) and the terms of how they are compared are simply and intuitively addressed, because of the simple use of iff. 4) By being simple and intuitive in definitions 1 and 2, I am able to very simply define finite (definition 3) and nonfinite (definition 4) sets, without any need to add anything, which are not already given by definitions 1 to 4. 5) Definition 5 very simply addresses nonstrict inequality as a range between strict inequality (< or >) and equality (=). You are invited to criticize what is written in this post and in http://www.internationalskeptics.com...9&postcount=15 post. Thank you. 
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30th June 2020, 10:03 AM  #32 
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You used a lot of words there. Did you have a point? My definition for relative cardinalities still stands: A <= B means/is defined as/if and only if there is an injection from A to B. ...and if you can't get along without supplemental information, no matter how obvious: ...where A and B are each sets, . is used to denote "cardinality of", and <= is the relationship that is the subject of the definition. 
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30th June 2020, 10:06 AM  #33 
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1st July 2020, 03:09 AM  #34 
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Let's see:
So, your definition is for: Relative measure. Relative cardinalities. Will you make up your mind? Mapping between sets' cardinalities, so you are using cardinality as a part of your definition of Cardinality. Moreover, you have said that you will provide me a definition of cardinality. Obvious? 
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1st July 2020, 08:23 AM  #35 
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1st July 2020, 07:25 PM  #36 
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Once again, you get things wrong. jsfisher is NOT "mapping between set's cardinality" so he is NOT using cardinality as a part of his definition. Moreover, is has provided you a definition of cardinality, multiple times.
Post #2: No. I have defined what the expression, A <= B, means. It means there exists an injection from A to B. Post #4: Cardinality is a relative measure of "size" of sets where A <= B if and only if there exists an injection from A to B. (quoted by you!) Post #14: Cardinality is a relative measure of "size" of sets where A <= B iff there exists an injection from A to B. (quoted by you, again) Post #20: A <= B <the thing being defined> if and only if <equivalent to saying "is defined as" or "means"> there is an injection from A to B. Post #21 is you misquoting (on purpose?) the definition given in post #20. Post #23 is again you misquoting (on purpose?) the definition given in post #20. Post #32: My definition for relative cardinalities still stands: A <= B means/is defined as/if and only if there is an injection from A to B. 
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1st July 2020, 11:45 PM  #37 
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By jsfisher's definition of Cardinality, A and B (which are actually the cardinalities of sets A and B) are inseparable parts of his definition, exactly as injection function from set A to set B is an inseparable part of his definition ,where this inseparability is based on IFF logical connective as as seen by its truth table, about the considered case, which is:
p="the cardinality of set A (notated as A) <= the cardinality of set B (notated as B))" q= "there is injection from set A to set B (notated as <= between the cardinality of set A (notated as A) and the cardinality of set B (notated as B)" Here is the iff truth table Code:
p iff q  F F T F T F T F F T T T And in all these times, cardinality is an inseparable part of his definition of Cardinality.  Also please see my post about relative measure in http://www.internationalskeptics.com...7&postcount=30 
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That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

2nd July 2020, 12:34 AM  #38 
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Meanwhile cardinality is an inseparable part of your definition of Cardinality (as shown in http://www.internationalskeptics.com...9&postcount=37) where this "definition" is a fundamental piece of your mathematical framework about sets.

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That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

2nd July 2020, 02:07 AM  #39 
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Who is the one that provides those semantics as parts of his posts about the definition of Cardinality? (the answer is given in http://www.internationalskeptics.com...5&postcount=34)

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2nd July 2020, 03:30 AM  #40 
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Ok let's reexamine what you wrote as follows:
An injection from set A to set B is a onetoone function from the members of set A to the members of set B, such that for every member of set A there is at most one matched member from set B, which provides at most two options by this relative measurement as follows: (The "size" of set A is relatively less than the size of set B) OR (The "size" of set A is relatively the same as the "size" of set B). Now comes the trick of "washingmachine of words to symbols" by replacing "the "size" of set A" with A symbol, and "the "size" of set B" with symbol B. Also by this trick of "washingmachine of words to symbols" "relatively less than" is replaced by <, and "relatively the same as the "size" of" is replaced by symbol =. And now after the "washingmachine of words to symbols" did its job, we can write: (A<B) OR (A=B) (also written as A<=B) means (written as iff logical connective) that there is injection from set A to set B. Now we take this "washingmachine of words to symbols" result and express it like this: Cardinality is a relative measure of "size" of sets where A <= B if and only if there exists an injection from A to B. (The meanings for strict equality and strict inequality of cardinalities follow directly.)  jsfisher, no matter how your "washingmachine of words to symbols" works, you are using cardinality as a part of your definition of Cardinality.  A more straightforward framework of this subject can be seen in http://www.internationalskeptics.com...8&postcount=31, where any criticism is welcome. Mathematics remains intact according to traditional mathematicians, if they take collections and their relations in terms of Plato philosophy school of thought. jsfisher, since you are using the term "intact" I believe that "set N is a complete whole" may be understood by you as "set N is intact". 
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That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

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